[0.7em]Flavor Covariant Formalism for Resonant Leptogenesis · Introduction Resonant Leptogenesis Resonant Leptogenesis For m N ˘ N, heavy Majorana neutrino self-energy effects on

Flavor Covariant Formalism forResonant Leptogenesis

P. S. BHUPAL DEV

Consortium for Fundamental Physics, University of Manchester, United Kingdom

based on

PSBD, P. Millington, A. Pilaftsis and D. Teresi,Nucl. Phys. B, in press [arXiv:1404.1003 [hep-ph]]

ICHEP 2014Valencia, Spain

Outline

Introduction

Flavor-Covariant Formalism

Rate Equations for Resonant Leptogenesis

Some Phenomenological Aspects

Conclusions

Bhupal Dev (Univ. Manchester) Flavor Covariant Formalism for Resonant Leptogenesis ICHEP 2014 1 / 14

Introduction Leptogenesis

Introduction to Leptogenesis

Leptogenesis: Lepton asymmetry from out-of-equilibrium decay of heavyMajorana neutrinos, converted into baryon asymmetry through (B + L)-violatingsphaleron interactions. [M. Fukugita and T. Yanagida, PLB 174, 45 (1986)]

• Resonant Leptogenesis

×NαNα

LCl

Φ†

(a)

×Nα Nβ

Φ

L LCl

Φ†

(b)

×Nα

L

Nβ

Φ†

LCl

Φ

(c)

Importance of self-energy effects (when |mN1 − mN2| ≪ mN1,2)[J. Liu, G. Segre, PRD48 (1993) 4609;

M. Flanz, E. Paschos, U. Sarkar, PLB345 (1995) 248;L. Covi, E. Roulet, F. Vissani, PLB384 (1996) 169;

...

J. R. Ellis, M. Raidal, T. Yanagida, PLB546 (2002) 228.]

Importance of the heavy-neutrino width effects: ΓNα

[A.P., PRD56 (1997) 5431; A.P. and T. Underwood, NPB692 (2004) 303.]

Warsaw, 22–27 June 2014 Flavour Covariance in Leptogenesis A. Pilaftsis

A cosmological consequence of the seesaw mechanism.

In ‘Vanilla’ Leptogenesis with hierarchical heavy neutrino masses(mN1 mN2 < mN3 ), a lower bound on mN1 & 109 GeV.[S. Davidson and A. Ibarra, PLB 535, 25 (2002); W. Buchmuller, P. Di Bari and M. Plumacher, NPB 643, 367 (2002)]

In conflict with gravitino overproduction bound: TR . 106 - 109 GeV.[see e.g., M. Kawasaki, K. Kohri, T. Moroi and A. Yotsuyanagi, PRD 78, 065011 (2008)]

Potential solution: Resonant Leptogenesis with ∆mN ∼ ΓN1,2 mN1,2 .[A. Pilaftsis, NPB 504, 61 (1997); PRD 56, 5431 (1997); A. Pilaftsis and T. Underwood, NPB 692, 303 (2004)]


Introduction Leptogenesis

Introduction to Leptogenesis

Leptogenesis: Lepton asymmetry from out-of-equilibrium decay of heavyMajorana neutrinos, converted into baryon asymmetry through (B + L)-violatingsphaleron interactions. [M. Fukugita and T. Yanagida, PLB 174, 45 (1986)]

• Resonant Leptogenesis

×NαNα

LCl

Φ†

(a)

×Nα Nβ

Φ

L LCl

Φ†

(b)

×Nα

L

Nβ

Φ†

LCl

Φ

(c)

Importance of self-energy effects (when |mN1 − mN2| ≪ mN1,2)[J. Liu, G. Segre, PRD48 (1993) 4609;

M. Flanz, E. Paschos, U. Sarkar, PLB345 (1995) 248;L. Covi, E. Roulet, F. Vissani, PLB384 (1996) 169;

...

J. R. Ellis, M. Raidal, T. Yanagida, PLB546 (2002) 228.]

Importance of the heavy-neutrino width effects: ΓNα

[A.P., PRD56 (1997) 5431; A.P. and T. Underwood, NPB692 (2004) 303.]

Warsaw, 22–27 June 2014 Flavour Covariance in Leptogenesis A. Pilaftsis

A cosmological consequence of the seesaw mechanism.

In ‘Vanilla’ Leptogenesis with hierarchical heavy neutrino masses(mN1 mN2 < mN3 ), a lower bound on mN1 & 109 GeV.[S. Davidson and A. Ibarra, PLB 535, 25 (2002); W. Buchmuller, P. Di Bari and M. Plumacher, NPB 643, 367 (2002)]

In conflict with gravitino overproduction bound: TR . 106 - 109 GeV.[see e.g., M. Kawasaki, K. Kohri, T. Moroi and A. Yotsuyanagi, PRD 78, 065011 (2008)]

Potential solution: Resonant Leptogenesis with ∆mN ∼ ΓN1,2 mN1,2 .[A. Pilaftsis, NPB 504, 61 (1997); PRD 56, 5431 (1997); A. Pilaftsis and T. Underwood, NPB 692, 303 (2004)]


Introduction Resonant Leptogenesis

Resonant Leptogenesis

For ∆mN ∼ ΓN , heavy Majorana neutrino self-energy effects on the leptonicCP-asymmetry become resonantly enhanced. [J. Liu and G. Segre, PRD 48, 4609 (1993); M. Flanz, E.

Paschos and U. Sarkar, PLB 345, 248 (1995); L. Covi, E. Roulet and F. Vissani, PLB 384, 169 (1996)]

Quasi-degeneracy can be obtained naturally from the approximate breaking ofsome leptonic symmetry in the Lagrangian

−LN = h αl LlΦ NR,α + NC

R,α [MN ]αβ NR,β + H.c.

An interesting RL scenario: Resonant `-genesis (RL`).Sphaleron processes preserve X` = B/3 − L` (with ` = e, µ, τ ). [J. Harvey and M. Turner, PRD 42,

3344 (1990); H. Dreiner and G. Ross, NPB 410, 188 (1993); J. Cline, K. Kainulainen and K. Olive, PRL 71, 2372 (1993)]

Baryon asymmetry can be generated in and protected by a single lepton flavor (`).[A. Pilaftsis, PRL 95, 081602 (2005)]

A minimal model of RL`: O(N)-symmetric heavy neutrino sector at some high-scale µX .Small mass splitting generated at low-scale due to RG effects:

MN = mN1 + ∆MN , where ∆MN = − mN

8π2ln(µX

mN

)Re[h†(µX) h(µX)]

[F. Deppisch and A. Pilaftsis, PRD 83, 076007 (2011)]]

A predictive RL model with testable consequences at energy frontier [PSBD, A. Pilaftsis, U.-k.

Yang, PRL 112, 081801 (2014)] and complementary effects at intensity frontier. [A. de Gouvea and P. Vogel,

PPNP 71, 75 (2013)]


Introduction Resonant Leptogenesis

Resonant Leptogenesis

For ∆mN ∼ ΓN , heavy Majorana neutrino self-energy effects on the leptonicCP-asymmetry become resonantly enhanced. [J. Liu and G. Segre, PRD 48, 4609 (1993); M. Flanz, E.

Paschos and U. Sarkar, PLB 345, 248 (1995); L. Covi, E. Roulet and F. Vissani, PLB 384, 169 (1996)]

Quasi-degeneracy can be obtained naturally from the approximate breaking ofsome leptonic symmetry in the Lagrangian



An interesting RL scenario: Resonant `-genesis (RL`).Sphaleron processes preserve X` = B/3 − L` (with ` = e, µ, τ ). [J. Harvey and M. Turner, PRD 42,

3344 (1990); H. Dreiner and G. Ross, NPB 410, 188 (1993); J. Cline, K. Kainulainen and K. Olive, PRL 71, 2372 (1993)]

Baryon asymmetry can be generated in and protected by a single lepton flavor (`).[A. Pilaftsis, PRL 95, 081602 (2005)]

A minimal model of RL`: O(N)-symmetric heavy neutrino sector at some high-scale µX .Small mass splitting generated at low-scale due to RG effects:

MN = mN1 + ∆MN , where ∆MN = − mN

8π2ln(µX

mN

)Re[h†(µX) h(µX)]

[F. Deppisch and A. Pilaftsis, PRD 83, 076007 (2011)]]

A predictive RL model with testable consequences at energy frontier [PSBD, A. Pilaftsis, U.-k.

Yang, PRL 112, 081801 (2014)] and complementary effects at intensity frontier. [A. de Gouvea and P. Vogel,

PPNP 71, 75 (2013)]


Introduction Flavor effects in RL

Flavordynamics of RL

Flavor effects are important in time-evolution of lepton asymmetry in RL models.Two sources of flavor effects, due to

Heavy neutrino Yukawa couplings h αl .[A. Pilaftsis, PRL 95, 081602 (2005); T. Endoh, T. Morozumi and Z.-h. Xiong, PTP 111, 123 (2004); P. Di Bari, NPB 727, 318 (2005);

S. Blanchet, P. Di Bari, D. A. Jones and L. Marzola, JCAP 1301, 041 (2013)]

Charged lepton Yukawa couplings y kl . [R. Barbieri, P. Creminelli, A. Strumia and N. Tetradis, NPB 575, 61 (2000); A.

Abada, S. Davidson, F. -X. Josse-Michaux, M. Losada and A. Riotto, JCAP 0604, 004 (2006); E. Nardi, Y. Nir, E. Roulet and J. Racker, JHEP

0601, 164 (2006); S. Blanchet and P. Di Bari, JCAP 0703, 018 (2007)]

Lead to three distinct physical phenomena: mixing, oscillation and (de)coherence.Fully flavor-covariant formalism essential to capture consistently all flavor effects.Flavor-diagonal Boltzmann equations:

nγHN

zdηNα

dz=

(1− ηN

α

ηNeq

)∑l

γNαLlΦ

nγHN

zdδηL

l

dz=∑α

(ηNα

ηNeq− 1)δγNα

LlΦ− 2

3δηL

l

∑k

[γLlΦ

LckΦ

c + γLlΦLkΦ

+ δηLk(γLkΦ

Lcl Φ

c − γLkΦLlΦ

)]Promote individual number densities to number density matrices in the so-called’density matrix’ formalism. [G. Sigl and G. Raffelt, NPB 406, 423 (1993)]

Obtain manifestly flavor-covariant transport equations.[PSBD, P. Millington, A. Pilaftsis and D. Teresi, NPB (2014)] (this talk)


Introduction Flavor effects in RL

Flavordynamics of RL

Flavor effects are important in time-evolution of lepton asymmetry in RL models.Two sources of flavor effects, due to

Heavy neutrino Yukawa couplings h αl .[A. Pilaftsis, PRL 95, 081602 (2005); T. Endoh, T. Morozumi and Z.-h. Xiong, PTP 111, 123 (2004); P. Di Bari, NPB 727, 318 (2005);

S. Blanchet, P. Di Bari, D. A. Jones and L. Marzola, JCAP 1301, 041 (2013)]

Charged lepton Yukawa couplings y kl . [R. Barbieri, P. Creminelli, A. Strumia and N. Tetradis, NPB 575, 61 (2000); A.

Abada, S. Davidson, F. -X. Josse-Michaux, M. Losada and A. Riotto, JCAP 0604, 004 (2006); E. Nardi, Y. Nir, E. Roulet and J. Racker, JHEP

0601, 164 (2006); S. Blanchet and P. Di Bari, JCAP 0703, 018 (2007)]

Lead to three distinct physical phenomena: mixing, oscillation and (de)coherence.Fully flavor-covariant formalism essential to capture consistently all flavor effects.Flavor-diagonal Boltzmann equations:

nγHN

zdηNα

dz=

(1− ηN

α

ηNeq

)∑l

γNαLlΦ

nγHN

zdδηL

l

dz=∑α

(ηNα

ηNeq− 1)δγNα

LlΦ− 2

3δηL

l

∑k

[γLlΦ

LckΦ

c + γLlΦLkΦ

+ δηLk(γLkΦ

Lcl Φ

c − γLkΦLlΦ

)]Promote individual number densities to number density matrices in the so-called’density matrix’ formalism. [G. Sigl and G. Raffelt, NPB 406, 423 (1993)]

Obtain manifestly flavor-covariant transport equations.[PSBD, P. Millington, A. Pilaftsis and D. Teresi, NPB (2014)] (this talk)


Flavor-Covariant Formalism Flavor-Covariant Theory

Flavor Covariant Formalism

Unitary flavor transformations:

Ll → V ml Lm L†,l → V l

m L†,m, NR,α → U βα NR,β , N†,αR → Uα

β N†,βR .

The Lagrangian



is invariant if h αl → V ml Uα

β h βm , [MN ]αβ → Uα

γ Uβδ [MN ]γδ.

Flavor-covariant quantization, e.g.

Ll(x) =

∫p,s

[(2EL(p))−

12

] i

l

([e−ip·x] j

i[u(p, s)] k

j bk(p, s) +[eip·x] j

i[v(p, s)] k

j d†k (p, s))

Matrix number densities: [nL] ml ∝ 〈b†,m bl〉, [nL] m

l ∝ 〈d†l dm〉, [nN ] βα ∝ 〈a†,β aα〉

Necessary to consider generalized discrete symmetries C,P, T, e.g.

dl = (bl)C ≡ G†,lm(bl)

C with G = V VT

Number densities transform as (nL)C = (nL)T, (nN)C = (nN)T(nN , nN not independent)

Define CP-“even” and CP-“odd” quantities:

nN =12

(nN + nN), δnN = nN − nN , δnL = nL − nL.







β N†,βR .

The Lagrangian





γ Uβδ [MN ]γδ.


Ll(x) =

∫p,s

[(2EL(p))−

12

] i

l

([e−ip·x] j

i[u(p, s)] k


i[v(p, s)] k

j d†k (p, s))


l ∝ 〈d†l dm〉, [nN ] βα ∝ 〈a†,β aα〉



C with G = V VT



nN =12








β N†,βR .

The Lagrangian





γ Uβδ [MN ]γδ.


Ll(x) =

∫p,s

[(2EL(p))−

12

] i

l

([e−ip·x] j

i[u(p, s)] k


i[v(p, s)] k

j d†k (p, s))


l ∝ 〈d†l dm〉, [nN ] βα ∝ 〈a†,β aα〉



C with G = V VT



nN =12



Flavor-Covariant Formalism Transport Equations

Flavor Covariant Transport Equations

nX(t) ≡ 〈nX (t; ti)〉t = Trρ(t; ti) nX (t; ti)

.

Markovian master equation for number density matrices:

ddt

nX(k, t) ' i〈 [HX0 , nX(k, t)] 〉t −

12

∫ +∞

−∞dt′ 〈 [Hint(t′), [Hint(t), nX(k, t)]] 〉t.

For charged-lepton and heavy-neutrino matrix number densities, we find:

ddt

[nLs1s2 (p, t)] m

l = − i[EL(p), nL

s1s2 (p, t)] m

l+ [CL

s1s2 (p, t)] ml

ddt

[nNr1r2 (k, t)] β

α = − i[EN(k), nN

r1r2 (k, t)] β

α+ [CN

r1r2 (k, t)] βα + Gαλ [C

Nr2r1 (k, t)] λ

µ Gµβ

Collision terms are of the form

[CLs1s2 (p, t)] m

l ⊃ −12

[Fs1s r1r2 (p, q, k, t)] n βl α [Γs s2r2r1 (p, q, k)] m α

n β ,

where F = nΦ nL ⊗(1− nN) − (

1 + nΦ) (

1− nL)⊗ nN are the statistical tensors, andΓ are the rank-4 absorptive rate tensors describing heavy neutrino decays and inversedecays.



Collision Rates for Decay and Inverse Decay

nΦ [nL] kl [γ(LΦ → N)] l β

k α

L

Φ

Nβ Nα

[hc] βk

[hc]lα

↓

Nβ(p, s)

Φ(q)

Lk(k, r)

[hc] βk

nΦ(q)[nLr (k)] kl Nα(p, s)

Φ(q)

Ll(k, r)

[hc]lα



Collision Rates for 2 ↔ 2 Scattering

nΦ [nL] kl [γ(LΦ → LΦ)] l n

k m

Φ

L

ΦLn Lm

hnβ h α

m

[hc] βk [hc]lα

Nβ Nα

↓

Nβ(p)

Φ(q2)

Ln(k2, r2)

Φ(q1)

Lk(k1, r1)

hnβ [hc] β

knΦ(q1)[n

Lr1(k1)]

kl

Nα(p)

Φ(q1)

Ll(k1, r1)

Φ(q2)

Lm(k2, r2)

[hc]lα h αm



Application to Resonant Leptogenesis

Classical statistics

Kinetic equilibrium

Degenerate spin degrees of freedom

Small deviation from equilibrium: [nL] ml + [nL] m

l ' 2 nLeq δ

ml

Unstable particle mixing accounted for by resummed Yukawa couplings:

h αl → h αl , [hc] αl . [A. Pilaftsis and T. Underwood, NPB 692, 303 (2004)]

Nα(p, s)

Φ(q)

Ll(k, r)

ε ε′[γN

LΦ]m β

l α ∝ hmαh βl + [hc]m

α[hc] βl

[δγNLΦ]

m β

l α ∝ hmαh βl − [hc]m

α[hc] βl

RIS-subtracted scattering rates: [γ′LΦLΦ ]

m kl n , [δγ

′LΦLΦ ]

m kl n , [γ

′LΦLcΦc ]

m kl n , [δγ

′LΦLcΦc ]

m kl n

Charged-lepton decoherence processes



Final Rate Equations

HN nγ

zd[ηN ] β

α

dz= − i

nγ

2

[EN , δη

N] β

α+[Re(γN

LΦ)] β

α− 1

2 ηNeq

ηN , Re(γN

LΦ) β

α

HN nγ

zd[δηN ] β

α

dz= − 2 i nγ

[EN , η

N] β

α+ 2 i

[Im(δγN

LΦ)] β

α− i

ηNeq

ηN , Im(δγN

LΦ) β

α

− 12 ηN

eq

δηN , Re(γN

LΦ) β

α

HN nγ

zd[δηL] m

l

dz= − [δγN

LΦ]m

l +[ηN ] α

β

ηNeq

[δγNLΦ]

m β

l α +[δηN ] α

β

2 ηNeq

[γNLΦ]

m β

l α

− 13

δηL, γLΦ

LcΦc + γLΦLΦ

m

l− 2

3[δηL]

nk

([γLΦ

LcΦc ]k m

n l − [γLΦLΦ ]

k mn l

)− 2

3

δηL, γdec

m

l+ [δγback

dec ] ml



Final Rate Equations: Mixing

HN nγ

zd[ηN ] β

α

dz= − i

nγ

2

[EN , δη

N] β

α+[Re(γN

LΦ)] β

α− 1

2 ηNeq

ηN , Re(γN

LΦ) β

α

HN nγ

zd[δηN ] β

α

dz= − 2 i nγ

[EN , η

N] β

α+ 2 i

[Im(δγN

LΦ)] β

α− i

ηNeq

ηN , Im(δγN

LΦ) β

α

− 12 ηN

eq

δηN , Re(γN

LΦ) β

α

HN nγ

zd[δηL] m

l

dz= − [δγN

LΦ]m

l +[ηN ] α

β

ηNeq

[δγNLΦ]

m β

l α +[δηN ] α

β

2 ηNeq

[γNLΦ]

m β

l α

− 13

δηL, γLΦ

LcΦc + γLΦLΦ

m

l− 2

3[δηL]

nk

([γLΦ

LcΦc ]k m

n l − [γLΦLΦ ]

k mn l

)− 2

3

δηL, γdec

m

l+ [δγback

dec ] ml



Final Rate Equations: Oscillation

HN nγ

zd[ηN ] β

α

dz= − i

nγ

2

[EN , δη

N] β

α+[Re(γN

LΦ)] β

α− 1

2 ηNeq

ηN , Re(γN

LΦ) β

α

HN nγ

zd[δηN ] β

α

dz= − 2 i nγ

[EN , η

N] β

α+ 2 i

[Im(δγN

LΦ)] β

α− i

ηNeq

ηN , Im(δγN

LΦ) β

α

− 12 ηN

eq

δηN , Re(γN

LΦ) β

α

HN nγ

zd[δηL] m

l

dz= − [δγN

LΦ]m

l +[ηN ] α

β

ηNeq

[δγNLΦ]

m β

l α +[δηN ] α

β

2 ηNeq

[γNLΦ]

m β

l α

− 13

δηL, γLΦ

LcΦc + γLΦLΦ

m

l− 2

3[δηL]

nk

([γLΦ

LcΦc ]k m

n l − [γLΦLΦ ]

k mn l

)− 2

3

δηL, γdec

m

l+ [δγback

dec ] ml

Notice:

O(h4) in total lepton asymmetry.Hence, different from ARSmechanism, which is an O(h6) effect.[E. Akhmedov, V. Rubakov, A. Smirnov, PRL 81, 1359 (1998);T. Asaka and M. Shaposhnikov, PLB 620, 17 (2005);B. Shuve and I. Yavin, arXiv:1401.2459 [hep-ph]]



Final Rate Equations: Oscillation

HN nγ

zd[ηN ] β

α

dz= − i

nγ

2

[EN , δη

N] β

α+[Re(γN

LΦ)] β

α− 1

2 ηNeq

ηN , Re(γN

LΦ) β

α

HN nγ

zd[δηN ] β

α

dz= − 2 i nγ

[EN , η

N] β

α+ 2 i

[Im(δγN

LΦ)] β

α− i

ηNeq

ηN , Im(δγN

LΦ) β

α

− 12 ηN

eq

δηN , Re(γN

LΦ) β

α

HN nγ

zd[δηL] m

l

dz= − [δγN

LΦ]m

l +[ηN ] α

β

ηNeq

[δγNLΦ]

m β

l α +[δηN ] α

β

2 ηNeq

[γNLΦ]

m β

l α

− 13

δηL, γLΦ

LcΦc + γLΦLΦ

m

l− 2

3[δηL]

nk

([γLΦ

LcΦc ]k m

n l − [γLΦLΦ ]

k mn l

)− 2

3

δηL, γdec

m

l+ [δγback

dec ] ml

Notice:

O(h4) in total lepton asymmetry.Hence, different from ARSmechanism, which is an O(h6) effect.[E. Akhmedov, V. Rubakov, A. Smirnov, PRL 81, 1359 (1998);T. Asaka and M. Shaposhnikov, PLB 620, 17 (2005);B. Shuve and I. Yavin, arXiv:1401.2459 [hep-ph]]



Final Rate Equations: Charged Lepton Decoherence

HN nγ

zd[ηN ] β

α

dz= − i

nγ

2

[EN , δη

N] β

α+[Re(γN

LΦ)] β

α− 1

2 ηNeq

ηN , Re(γN

LΦ) β

α

HN nγ

zd[δηN ] β

α

dz= − 2 i nγ

[EN , η

N] β

α+ 2 i

[Im(δγN

LΦ)] β

α− i

ηNeq

ηN , Im(δγN

LΦ) β

α

− 12 ηN

eq

δηN , Re(γN

LΦ) β

α

HN nγ

zd[δηL] m

l

dz= − [δγN

LΦ]m

l +[ηN ] α

β

ηNeq

[δγNLΦ]

m β

l α +[δηN ] α

β

2 ηNeq

[γNLΦ]

m β

l α

− 13

δηL, γLΦ

LcΦc + γLΦLΦ

m

l− 2

3[δηL]

nk

([γLΦ

LcΦc ]k m

n l − [γLΦLΦ ]

k mn l

)− 2

3

δηL, γdec

m

l+ [δγback

dec ] ml

Captures three distinct physical phenomena: mixing, oscillation and decoherence.



Final Rate Equations: Charged Lepton Decoherence

HN nγ

zd[ηN ] β

α

dz= − i

nγ

2

[EN , δη

N] β

α+[Re(γN

LΦ)] β

α− 1

2 ηNeq

ηN , Re(γN

LΦ) β

α

HN nγ

zd[δηN ] β

α

dz= − 2 i nγ

[EN , η

N] β

α+ 2 i

[Im(δγN

LΦ)] β

α− i

ηNeq

ηN , Im(δγN

LΦ) β

α

− 12 ηN

eq

δηN , Re(γN

LΦ) β

α

HN nγ

zd[δηL] m

l

dz= − [δγN

LΦ]m

l +[ηN ] α

β

ηNeq

[δγNLΦ]

m β

l α +[δηN ] α

β

2 ηNeq

[γNLΦ]

m β

l α

− 13

δηL, γLΦ

LcΦc + γLΦLΦ

m

l− 2

3[δηL]

nk

([γLΦ

LcΦc ]k m

n l − [γLΦLΦ ]

k mn l

)− 2

3

δηL, γdec

m

l+ [δγback

dec ] ml

Captures three distinct physical phenomena: mixing, oscillation and decoherence.


Some Phenomenological Aspects Minimal Model of RLτ

Minimal Model of Resonant τ -Genesis

Consider a minimal model of Resonant τ -Genesis. [A. Pilaftsis, PRL 95, 081602 (2005)]

Yukawa couplings break the O(3)-symmetry to (almost) U(1)Le+Lµ × U(1)Lτ :

h =

0 ae−iπ/4 aeiπ/4

0 be−iπ/4 beiπ/4

0 0 0

+δh, where δh =

εe 0 0εµ 0 0

ετ κ1e−i(π4 −γ1) κ2ei(π4 −γ2)

From type-I seesaw formula: Mν ' − v2

2 h M−1N hT = 0 in the limit δh→ 0.

To satisfy neutrino oscillation data, require

a2 =2mN

v2κN

(Mν,11 −

M2ν,13

Mν,33

), b2 =

2mN

v2κN

(Mν,22 −

M2ν,23

Mν,33

),

ε2e =

2mN

v2

M2ν,13

Mν,33, ε2

µ =2mN

v2

M2ν,23

Mν,33, ε2

τ =2mN

v2 Mν,33 .

where κN ≡ 18π2 ln

(µXmN

) [2κ1κ2 sin(γ1 + γ2) + i(κ2

2 − κ21)].

Input model parameters: mN , κ1, κ2, γ1, γ2.


Some Phenomenological Aspects Minimal Model of RLτ

Benchmark Points

Parameters BP1 BP2 BP3

mN 120 GeV 400 GeV 5 TeVγ1 π/4 π/3 3π/8γ2 0 0 π/2κ1 4× 10−5 2.4× 10−5 2× 10−4

κ2 2× 10−4 6× 10−5 2× 10−5

a (7.41− 5.54 i)× 10−4 (4.93− 2.32 i)× 10−3 (4.67 + 4.33 i)× 10−3

b (1.19− 0.89 i)× 10−3 (8.04− 3.79 i)× 10−3 (7.53 + 6.97 i)× 10−3

εe 3.31× 10−8 5.73× 10−8 2.14× 10−7

εµ 2.33× 10−7 4.3× 10−7 1.5× 10−6

ετ 3.5× 10−7 6.39× 10−7 2.26× 10−6

Observable BP1 BP2 BP3 Exp. Limit

BR(µ→ eγ) 4.5× 10−15 1.9× 10−13 2.3× 10−17 < 5.7× 10−13

BR(τ → µγ) 1.2× 10−17 1.6× 10−18 8.1× 10−22 < 4.4× 10−8

BR(τ → eγ) 4.6× 10−18 5.9× 10−19 3.1× 10−22 < 3.3× 10−8

BR(µ→ 3e) 1.5× 10−16 9.3× 10−15 4.9× 10−18 < 1.0× 10−12

RTiµ→e 2.4× 10−14 2.9× 10−13 2.3× 10−20 < 6.1× 10−13

RAuµ→e 3.1× 10−14 3.2× 10−13 5.0× 10−18 < 7.0× 10−13

RPbµ→e 2.3× 10−14 2.2× 10−13 4.3× 10−18 < 4.6× 10−11

|Ω|eµ 5.8× 10−6 1.8× 10−5 1.6× 10−7 < 7.0× 10−5

〈m〉 [eV] 3.8× 10−3 3.8× 10−3 3.8× 10−3 < (0.11–0.25)


Some Phenomenological Aspects Numerical Results

Lepton Asymmetry for BP 2

-∆ΗL

+∆ΗL

-∆Ηobs

L

mN = 400 GeV

zc0.2 1 10 2010

-9

10-8

10-7

10-6

10-5

z = mNT

±∆

ΗL

total, Ηin

N =0, ∆Ηin

L =I

total, Ηin

N =0, ∆Ηin

L =0

total, Ηin

N =Ηeq

N, ∆Η

in

L =0

N diag., Ηin

N =Ηeq

N, ∆Η

in

L =0

N diag., analytic

L diag., Ηin

N =Ηeq

N, ∆Η

in

L =0

N, L diag., Ηin

N =Ηeq

N, ∆Η

in

L =0


Conclusions

Conclusions

A fully flavor-covariant formalism essential to describe consistently all flavoreffects.

Main new ingredients: rank-4 rate tensors, interpreted as unitarity cuts of partialthermal self-energies.

A complete and unified description of Resonant Leptogenesis, capturing threephysically distinct phenomena:

resonant mixing between heavy neutrinos,coherent oscillations between heavy-neutrino flavors,quantum decoherence effects in the charged-lepton sector.

As an application, we discussed some numerical results for a minimal model ofresonant τ -Genesis.

Showed that the final lepton asymmetry can be enhanced up to an order ofmagnitude, for mN ∼ 200 - 1000 GeV.

For more details, see our paper arXiv:1404.1003 [hep-ph] (109 pages!).


Backup slides

Backup slides

BP 2: Lepton Flavor Content

-∆Ηobs

L

mN = 400 GeV

zc0.2 1 10 2010

-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

z = mNT

È∆ΗlmL

È -∆ΗΤΤ

L

+∆ΗΜΜ

L

-∆Ηee

L

È∆ΗΤΜ

L ÈÈ∆Η

Τe

L ÈÈ∆Η

Μe

L È


Backup slides

BP 2: Heavy Neutrino Flavor Content

mN = 400 GeV

zc0.2 1 10 2010-14

10-12

10-10

10-8

10-6

10-4

10-2

100

ÈΗ ΑΒ

NΗ eqN

-∆

ΑΒ

È

Η11N Ηeq

N - 1

Η22N Ηeq

N - 1

Η33N Ηeq

N - 1

ÈΗ12N Ηeq

N ÈÈΗ13

N ΗeqN È

ÈΗ23N Ηeq

N È


Backup slides

BP 1

-∆ΗL

+∆ΗL

-∆Ηobs

L

mN = 120 GeV

zc0.2 1 10 2010

-9

10-8

10-7

10-6

10-5

10-4

z = mNT

±∆

ΗL

total, Ηin

N =0, ∆Ηin

L =I

total, Ηin

N =0, ∆Ηin

L =0

total, Ηin

N =Ηeq

N, ∆Η

in

L =0

N diag., Ηin

N =Ηeq

N, ∆Η

in

L =0

N diag., analytic

L diag., Ηin

N =Ηeq

N, ∆Η

in

L =0

N, L diag., Ηin

N =Ηeq

N, ∆Η

in

L =0


Backup slides

BP 3

-∆ΗL

+∆ΗL

-∆Ηobs

L

0.2 1 10 2010

-8

10-7

10-6

10-5

10-4

10-3

z = mNT

±∆

ΗL

mN = 5 TeV total, Ηin

N =0, ∆Ηin

L =I

total, Ηin

N =0, ∆Ηin

L =0

total, Ηin

N =Ηeq

N, ∆Η

in

L =0

N diag., Ηin

N =Ηeq

N, ∆Η

in

L =0

N diag., analytic

L diag., Ηin

N =Ηeq

N, ∆Η

in

L =0

N, L diag., Ηin

N =Ηeq

N, ∆Η

in

L =0


Documents

[0.7em]Flavor Covariant Formalism for Resonant Leptogenesis · Introduction Resonant Leptogenesis Resonant Leptogenesis For m N ˘ N, heavy Majorana neutrino self-energy effects on